Balking and reneging in an M/M/s system. Suppose that X(t) is an M/M/s queueing process with arrival rate λ and service rate μ. Consider the variation in which items balk at entering, such that when n...

Balking and reneging in an M/M/s system. Suppose that X(t) is an M/M/s queueing process with arrival rate λ and service rate μ. Consider the variation in which items balk at entering, such that when n items are in the system, the arrival rate is biλ. For instance, bi could be the probability that an arrival decides to enter the system when i items are in the system (e.g., bi = 1(i ≤ K)). Assume bi = e−αi/μ, where i/μ is an estimate for the average waiting time when i items are present and α > 0. Specify the transition rates for the system, give a criterion for the process to be ergodic, and determine its stationary distribution.